Bounds on One-Shot Information Transmission Capacities of Discrete-Time Quantum Markov Semigroups

المفاهيم الأساسية

The authors derive upper and lower bounds on the one-shot ε-error information transmission capacities (classical, quantum, entanglement-assisted classical, and private classical) of a discrete-time quantum Markov semigroup in terms of the structure of the peripheral space of the underlying quantum channel.

الملخص

The key highlights and insights of the content are:

1. The authors consider an open quantum system undergoing Markovian dynamics, modeled by a discrete-time quantum Markov semigroup {Φn}n∈N, where Φ is a quantum channel and n is the discrete time parameter.

2. They analyze the one-shot ε-error information transmission capacities of Φn for finite time n ∈ N and error ε ∈ [0, 1), focusing on the transmission of classical, quantum, and private classical information, as well as entanglement-assisted classical communication.

3. The authors derive upper and lower bounds on these capacities in terms of the structure of the peripheral space of the channel Φ, which is characterized by the block dimensions dk = dim Hk,1 and the states δk.

4. The achievability bounds are obtained by constructing explicit communication protocols that work with zero error, while the converse bounds use techniques based on the max-relative entropy and the ε-hypothesis testing relative entropy.

5. In the asymptotic limit as n → ∞, the authors provide the corresponding bounds on the capacities, showing that they depend only on the peripheral space structure of the channel Φ.

6. The results generalize and improve upon previous work on the asymptotic capacities of discrete-time quantum Markov semigroups.

الإحصائيات

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اقتباسات

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الرؤى الأساسية المستخلصة من

Information transmission under Markovian noise

by Satvik Singh... في arxiv.org 09-27-2024

https://arxiv.org/pdf/2409.17743.pdf

استفسارات أعمق

How do the bounds derived in this work compare to the average-error capacities studied in previous literature?

The bounds derived in this work for the one-shot ε-error capacities of discrete-time quantum Markov semigroups (dQMS) are primarily focused on the worst-case error scenario, contrasting with the average-error capacities often studied in previous literature. In the context of quantum Shannon theory, average-error capacities are typically evaluated under the assumption that the error vanishes in the asymptotic limit, allowing for a more relaxed error criterion. The results presented in this paper, particularly in Theorem 3.1, provide both upper and lower bounds that are specifically tailored for finite n and a non-zero error ε, which is a more realistic scenario for practical applications.
While previous studies, such as those referenced in the paper ([FRT24], [SRD24]), have established bounds for average-error capacities, the current work emphasizes the importance of understanding the behavior of quantum channels under sequential uses, where the error can be non-negligible. This focus on one-shot capacities allows for a more nuanced understanding of the information transmission capabilities of quantum channels in practical settings, where the assumption of asymptotic behavior may not hold.

Can the achievability bounds be improved by explicitly taking the error ε into account, or are the current bounds tight?

The achievability bounds presented in Theorem 3.1 are derived under the assumption of zero error (ε = 0) and subsequently generalized to accommodate a non-zero error ε. However, the authors note that it remains unclear whether these bounds can be improved by explicitly incorporating ε into the analysis. The existing bounds are based on protocols that work perfectly in the zero-error case, and while they provide a solid foundation, they may not fully capture the complexities introduced by non-zero error probabilities.
In comparison, the bounds established in previous works, such as those in [FRT24], do take ε into account and yield slightly better results for the quantum and classical capacities. This suggests that there is potential for refinement in the current bounds by developing protocols that explicitly consider the error ε during the encoding and decoding processes. Therefore, while the current bounds are significant, they may not be tight, and further exploration into the role of ε could lead to improved achievability bounds.

What are the implications of these results for practical applications of discrete-time quantum Markov semigroups in quantum communication and information processing?

The results of this study have profound implications for practical applications in quantum communication and information processing, particularly in the context of discrete-time quantum Markov semigroups (dQMS). By establishing clear upper and lower bounds on the one-shot ε-error capacities, the findings provide a framework for evaluating the performance of quantum channels under realistic conditions where errors cannot be ignored.
In practical terms, these results can guide the design of quantum communication protocols that are robust against noise and errors, which are inherent in real-world quantum systems. The ability to quantify the capacities for classical, quantum, entanglement-assisted classical, and private classical communication allows for the optimization of information transmission strategies tailored to specific applications, such as secure communication and quantum cryptography.
Moreover, the focus on sequential uses of quantum channels reflects the operational realities of many quantum information tasks, where resources are often limited, and the number of channel uses is finite. This work encourages further research into the dynamics of open quantum systems and their interaction with Markovian noise, ultimately contributing to the development of more efficient quantum technologies and enhancing the reliability of quantum information processing systems.